Given a flow
θ
g
,
g
∈
G
{\theta _g},g \in G
a group, over a probability space
(
Ω
,
F
,
P
)
(\Omega ,\mathfrak {F},P)
and a
G
G
-valued random variable
Z
Z
, we exhibit the Lebesgue decomposition of the measure
P
∘
θ
Z
−
1
P \circ \theta _Z^{ - 1}
relative to
P
P
, and give necessary and sufficient conditions for equality
(
P
∘
θ
Z
−
1
=
P
)
(P \circ \theta _Z^{ - 1} = P)
, absolute continuity
(
P
∘
θ
Z
−
1
≪
P
)
(P \circ \theta _Z^{ - 1} \ll P)
, and singularity
(
P
∘
θ
Z
−
1
⊥
P
)
(P \circ \theta _Z^{ - 1} \bot P)
in terms of the Haar measure. The proof rests on the theory of “Palm measures” as developed by Mecke and the authors. Specializing the group
G
G
, we retrieve some known results for the integers and real line, and compute the Radon-Nikodým derivatives in various cases.